Nonlinear Dyn (2017) 88:1691–1705 DOI 10.1007/s11071-017-3339-y ORIGINAL PAPER Solving Painlevé paradox: (P–R) sliding robot case Hesham A. Elkaranshawy · Khaled T. Mohamed · Ahmed S. Ashour · Hassan M. Alkomy Received: 9 July 2016 / Accepted: 6 January 2017 / Published online: 26 January 2017 © Springer Science+Business Media Dordrecht 2017 Abstract For a rigid body or a multibody system sliding on a rough surface, a range of uncertainty or non-uniqueness of solution could be found, which is termed: Painlevé paradox. Painlevé paradox is the rea- son of a wide range of undesired bouncing motions which are observed during sliding. As Painlevé para- dox is a practical problem in case of multibody systems, this research work has investigated that paradox. In this research work, the condition leading to Painlevé para- dox has been determined for a general multibody sys- tem. Investigating the motion of a prismatic–revolute (P–R), sliding robot has been conducted. In order to solve the paradox and ﬁnd the motion, a tangential impact is assumed at the contact point. The impact model has been developed and the paradox, conse- quently, has been solved. Consequently, the kinematics of the motion has been speciﬁed. Keywords Painlevé paradox · Tangential impact · Friction · Multibody systems · Robot dynamics H. A. Elkaranshawy · A. S. Ashour · H. M. Alkomy (B ) Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Alexandria, Egypt e-mail: eng.alkomy@gmail.com; h.m.alkomy@alexu.edu.eg K. T. Mohamed Department of Mechanical Engineering, Faculty of Engineering, Alexandria University, Alexandria, Egypt 1 Introduction The motion of multibody systems may face many dynamical problems and obstacles which affect their functional operation. These problems may occur due to several reasons; geometrical reasons, force-related reasons, material reasons… etc. These dynamical prob- lems could inﬂuence the motion dramatically. Self- locking for some parallel robots is one of these dynam- ical problems that may lead to either a sudden stop during the robot’s motion or even inability to start the motion [1]. One of these problems, which has deep effect on the motion of sliding robots, is the undesired bouncing motion. Good understanding of this prob- lem and modeling it, using the classical mechanics concepts, are so important for the dynamical analy- sis and control issues. Pushing a chalk on a blackboard is one of the simplest and best examples of the unde- sired bouncing motions. Sometimes during the push- ing process, the chalk leaves the blackboard and shows detachments [2]. The cause of bouncing motion may be some external forces and inertia actions or may be due to the conﬁguration and contact parameters such as lengths, masses and/or the coefﬁcient of friction. Liu et al. [3] discussed the difference between these two main sources of bouncing motion and found many valuable results. In this research, we devote ourselves to study the bouncing motion in robotic systems related to the conﬁguration and contact parameters, as one of the common problems that needs intensive research. Exactly, we study the bouncing motion that occurs for 123